Solution Found!
Prove that for the system dx dt = F( x, y), dy dt = G( x, y) there is at most one
Chapter 9, Problem 22(choose chapter or problem)
Prove that for the system dx dt = F( x, y), dy dt = G( x, y) there is at most one trajectory passing through a given point ( x0, y0). Hint: Let C0 be the trajectory generated by the solution x = 0(t), y = 0(t), with 0(t0) = x0, 0(t0) = y0, and let C1 be the trajectory generated by the solution x = 1(t), y = 1(t), with 1(t1) = x0, 1(t1) = y0. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that C0 and C1 are the same. 2
Questions & Answers
QUESTION:
Prove that for the system dx dt = F( x, y), dy dt = G( x, y) there is at most one trajectory passing through a given point ( x0, y0). Hint: Let C0 be the trajectory generated by the solution x = 0(t), y = 0(t), with 0(t0) = x0, 0(t0) = y0, and let C1 be the trajectory generated by the solution x = 1(t), y = 1(t), with 1(t1) = x0, 1(t1) = y0. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that C0 and C1 are the same. 2
ANSWER:Step 1 of 5
It is to be shown that for the system
There is at most one trajectory passing through a given point 
Let 
With
